Discontinuous Extraction of a Nonrenewable Resource
by
Eric Iksoon Im1
Department of Economics,
College of Business & Economics, University of Hawaii at Hilo, Hilo, Hawaii
Ujjayant Chakravorty
Department of Economics, Emory University, Atlanta, Georgia
James Roumasset
Department of Economics, University of Hawaii at Manoa, Honolulu, Hawaii
Abstract
This paper examines the optimal extraction sequence of nonrenewable resources in the
presence of multiple demands. We provide conditions under which extraction of a
nonrenewable resource may be discontinuous over the course of its depletion.
JEL classification: Q3; Q4
Keywords: Backstop technology; Dynamic optimization; Energy resources; Herfindahl
principle; Multiple demands
This Version: May 2004
1
Eric Iksoon Im, 200 W. Kawili Steet, Hilo, Hawaii 96720-4091; ph: (808)974-7462; fax:
(808)974-7685; email: eim@hawaii.edu.
1
1. Introduction
A fundamental theorem of resource economics suggests that extraction of identical
deposits of a nonrenewable resource should be in the order of their cost of extraction
(e.g., Herfindahl (1967), Solow and Wan (1976), and Lewis (1982)). Gaudet, Moreaux
and Salant (2001), use a model of trash hauling between cities (in our case, demands) and
landfills with a fixed capacity (analogous to resources), to suggest that in the presence of
setup costs a city may temporarily abandon a low marginal cost site (and use a higher
cost one) and return to the former at a later date. An implication of their result is that a
nonrenewable resource may be extracted discontinuously, i.e., over two disjoint time
periods. In this paper, we show that discontinuous extraction of a nonrenewable resource
is still possible across demands, even without setup costs. That is, a resource may be used
in a particular demand, then abandoned for a time, only to be used in another demand
later in time. We provide conditions for this phenomenon to occur.
We modify the framework of dynamic optimization in Chakravorty and Krulce (1994,
henceforth CK) who consider two nonrenewable resources, oil (O) and coal (C), for two
demands, electricity (E) and transportation (T), by adding a third backstop resource (B)
with an infinite supply (e.g., solar power).2 While the assumption of a constant unit
extraction cost in CK is retained for each resource (ci, i = O , C , B ), we specify conversion
costs as both resource and energy specific ( z ij , i = O, C , B; j = E , T ) so that the net cost
of resource i for demand j is wij = ci + z ij .
The planner chooses instantaneous extraction rates of resource i for demand j, qij (t ) ,
which maximizes the discounted social surplus, W:


∑ qij
W = ∫ e − rt ∑  ∫ i D −j 1 ( x)dx  − ∑ (ci + z ij )q ij (t ) − ∑ λi (t )∑ qij (t ) dt
0
 i, j
i
j
0
 j 

∞
subject to
2
At least three resources are needed for discontinuous extraction, which is also the case in Gaudet et al.
2
qij (t ) ≥ 0; Qi (t ) ≥ 0; Q& i (t ) = −∑ qij (t )
j
where r denotes the discount rate, D −j 1 the inverse energy demand function for demand j,
Qi (t ) the initial stock (assumed known) of resource i, and λi (t ) the co-state variable for
resource i. Define the resource price for demand j as p j (t ) = D −j 1 (∑ qij (t )) and the price
i
of resource i for demand j as p ij (t ) = ci + z ij + λi (t ) ≡ wij + λi (t ) . The necessary and
sufficient conditions3 are
Q& i (t ) = −∑ qij (t )
(1)
λ&i (t ) = rλi (t )
(2)
p j (t ) ≤ pij (t ) (if < then qij (t ) = 0 )
(3)
lim e − rt λi (t ) ≥ 0 ; lim e − rt λi (t ) Qi (t ) = 0
(4)
j
t →∞
t →∞
where (2) implies λi (t ) = λi (0) e rt . Substituting (2) into (4) for nonrenewable resource i (
= C, O ), we obtain lim e − rt (λi (0) e rt ) Qi (t ) = λi (0) lim Qi (t ) which gives lim Qi (t ) = 0 .
t →∞
t →∞
t →∞
For the backstop resource ( i = B ) which is in infinite supply, Q B (t ) > 0 for all t, so that
(
)
lim e − rt λ B (0) e rt Q B (t ) = λ B (0) lim QB (t ) which yields λ B (0) = 0 , hence λ B (t ) = 0 .
t →∞
t →∞
2. Optimal Extraction Sequence
Let us consider the special case in which oil is the cheapest resource for all demands and
the backstop is the most expensive:
Assumption: 0 < wOj < wCj < wBj < ∞ .
3
The proof of sufficiency is essentially the same as in CK, hence suppressed.
3
Then, as shown by Chakravorty, Krulce, and Roumasset (2003), the ordering of the
shadow prices is exactly the reverse of the ordering of net costs wij , i.e.,
λO (t ) > λC (t ) > λ B (t ) = 0 . Note that since both oil and coal are nonrenewable resources,
they will eventually be exhausted and the backstop will be used for both demands. By
Proposition 7 of their paper, the order of extraction in each demand must be in the order
of the net costs (OCB), i.e., oil followed by coal and then by the backstop (as illustrated
in Fig. 1), although not every resource need be extracted for each demand.
3. Conditions for Discontinuous Resource Extraction
In this section, we demonstrate graphically the possibility of discontinuous extraction of a
nonrenewable resource, and then provide necessary and sufficient conditions for the
discontinuity to occur. In Fig.1, the (energy) resource price for each demand is depicted
as an envelope curve: in bold solid for transportation and in bold dash for electricity.
Note that coal is extracted in phase II, and again in phase IV. There is no extraction of
coal in the intermediate phase III.
<Figure 1 here>
The switch point sequence for Fig. 1 is S1: 0 < t1E < t 2 E < t1T <t 2T < ∞ . Note that if the
pOE (t ) curve in the lower part of Fig. 1 shifts up, oil may not be used in electricity, but
coal extraction will remain discontinuous but with an altered switch point sequence S2:
t1E ≤ 0 < t 2 E < t1T <t 2T < ∞ . Either S1 or S2 is equivalent to the following three
inequalities:
(i ). 0 < t 2 E ; (ii ). t 2 E < t1T ; (iii ). t1 j < t 2 j ,
j = E, T .
(5)
4
Under these two sequences, coal is extracted first for electricity and then for
transportation. We can now state
PROPOSITION 1: Coal is extracted discontinuously, first for electricity (E) and then
for transportation (T) after a time delay, iff
(a). λC (0) < wBE − wCE ;
 wCj − wOj  λO (0)
w − wOT
(b). 1 + max 
,
< 1 + CT
<
wBE − wCE
 wBj − wCj  λC (0)
j = E, T . 4
Proof: At the switch points for demand j, pOj = pCj and pCj = p Bj , which gives:
wCj − wOj
1
1 wBj − wCj
t1 j = ln
.
; t 2 j = ln
r λO (0) − λC (0)
r
λC (0)
(6)
In view of (5), it suffices to show that (i), (ii) and (iii) are equivalent to conditions (a) and
(b).
Since 0 < λC (0) < λO (0) < ∞ in light of the Assumption in Section 2,
(i ). 0 < t 2 E
⇔ (a). λ C (0) < wBE − wCE ;
(ii ). t 2 E < t1T
⇔ (b1).
4
λ O ( 0)
w − wOT
;
< 1 + CT
λ C ( 0)
wBE − wCE
There exists a subset of w = ( wOE , wCE , wBE , wOT , wCT , wBT ) which admits conditions (a) and (b) ,
e.g., w = ( 4,5,6,1,4,6) . Given w ,
λO (0)
and
λC (0)
still depend on other factors such as the initial
stocks of resources, the discount rate and the magnitude of demands, hence are not determined solely by
w.
5
(iii ). t1 j < t 2 j
wCj − wOj
⇔
λO (0) − λC (0)
<
wBj − wCj
λC (0)
⇔ (b2).
 wCj − wOj 
λO (0)
> 1 + max 
.
λC (0)
 wBj − wCj 
Noting that (b1) and (b2) jointly are equivalent to (b) in Proposition 1 completes the
proof.
Q.E.D.
The conditions in Proposition 1 can be re-expressed as three simple inequality constraints
for λC (0) and λC (0) : λC (0) < α , λO (0) < β λC (0) , and λO (0) > γ λC (0) where
α ≡ wBE − wCE > 0 ;
β ≡ 1+
wCT − wOT
> 1 ; and
wBE − wCE
 wCj − wOj 
 > 1,
 wBj − wCj 
γ ≡ 1 + max 
which are graphed in Fig. 2. The entire grey area represents an open set of
{((λC (0), λO (0)) | 0 < λC (0), λO (0) < ∞}
which satisfies the conditions for either S1 or S2
to occur. The only difference between the two sequences is that 0 < t1E for S1 and
t1E ≤ 0 for S2.
Setting t1E = 0 in (6), we obtain λO (0) = µ + λC (0) where
µ ≡ wCE − wOE > 0 . Hence, we can restate the difference in terms of λC (0) and λO (0) :
λO (0) > µ + λC (0) for S1 and λO (0) < µ + λC (0) for S2. In Fig. 2, the line
λO (0) = µ + λC (0) splits the grey area into two. The dark grey area, an open set, is the
domain of (λC (0), λO (0) ) which admits sequence S1, and the light grey area which is
inclusive of the splitting line is the domain of (λC (0), λO (0) ) for sequence S2. If
µ ≥ µ * ≡ α ( β − 1) , there exists no (λC (0), λO (0) ) which admits sequence S1, and the
entire grey area is the domain for sequence S2.
6
<Figure 2 here>
Note that Fig. 1 is symmetric with respect to E and T. That is, if E and T were
interchanged, coal would still be extracted discontinuously with the altered switch point
sequences S1′ : 0 < t1T < t 2T < t1E <t 2 E < ∞ and S 2′ :
t1T ≤ 0 < t 2T < t1E <t 2 E < ∞ which
are identical to S1 and S2. Proposition 1 will continue to hold in this case. Since {S1, S2,
S1′ , S 2′ } is the complete set of switch point sequences which admits discontinuous coal
extraction, we may then generalize Proposition 1 (without proof) as
PROPOSITION 2 (General): Coal is extracted discontinuously iff
(a) λC (0) < wBj − wCj ;
wCj * − wOj *
 w − wOh  λO (0)
< 1+
(b) 1 + max  Ch
<
wBj − wCj
 wBh − wCh  λC (0)
( j , j * = E or T ; j ≠ j * ; h = j , j * ) .
4. Conclusion
This paper provides conditions under which a nonrenewable resource can be extracted
discontinuously in a model with two resources and a renewable backstop resource. Given
that the Herfindahl principle (least cost first) must be preserved within each demand, a
total of three resources are necessary for the discontinuous extraction to occur. In a
typical economy with multiple demands for resources with different grades, an observed
pattern of discontinuous resource extraction may appear chaotic, but may actually be a
result of optimal behavior as suggested in this paper.
7
References
Chakravorty, U. and D. L. Krulce, 1994, “Heterogeneous Demand and Order of Resource
Extraction,” Econometrica 62, 1445-1452.
Chakravorty, U., D.L. Krulce, and J. Roumasset, 2003, “Specialization and
Nonrenewable Resources: Ricardo meets Ricardo,” (Unpublished), Department of
Economics, Emory University.
Gaudet, G., Moreaux, M., Salant, S., 2001, “Intertemporal Depletion of Resource Sites
by Spatially Distributed Users,” American Economic Review 91 (4), 1149-59.
Herfindahl, O.C., 1967, “Depletion and Economic Theory,” in M. Gaffney, ed.,
Extractive Resources and Taxation, (University of Wisconsin Press), 63-90.
Lewis, T. R., 1982, “Sufficient Conditions for Extracting Least Cost Resource First,”
Econometrica 50, 1081-1083.
Solow R., Wan, F.Y., 1976, “Extraction Costs in the Theory of Exhaustible Resources,”
The Bell Journal of Economics 7, 359-370.
8
pT (t ) : Energy price for transportation
p E (t ) : Energy price for electricity
pOT (t )
pCT (t )
p B T (t )
backstop
coal
oil
pCE (t )
pOE (t )
p B E (t )
backstop
coal
oil
t1 E
0
I
t2E
II
t 2T
t1T
III
IV
t
V
Fig. 1: Discontinuous Coal Extraction: Coal is extracted in phase II and IV, but not in III.
9
λO (0)
µ * + λC (0)
β λC (0)
µ + λC (0)
45 0
γ λC (0)
µ*
µ
λC (0) = α
0
α
Fig. 2: Coal Extraction is Discontinuous in the Shaded Open Set
λC (0)